Torus
A torus (plural: tori or toruses) is a tube shape that looks like a doughnut or an inner tube. In geometry, a torus is made by rotating a circle in three dimensional space. To make a torus, the circle is rotated around a line (called the axis of rotation) that is in the same plane as the circle. Usually, the line does not touch the circle, so the torus has a hole through the center, and the torus is called a ring torus. For a ring torus, the axis of rotation passes through the center of the hole.
In topology, sizes don't matter, and a torus is any shape that has one hole through it.
The ring torus is the most well-known type of torus, but other types exist. If the line the circle rotates around is tangent to the circle, then it becomes a horn torus, and if it passes through the circle then it is a spindle torus. A toroid is a surface made by rotating any shape around a line, so a torus is one kind of toroid.
If the torus is filled to make a solid shape, it is called a solid torus. A solid torus is often simply called a torus. A solid torus is made by rotating a disk (a filled-in circle) around a line. Common objects that have the shape of a solid torus are a doughnut, a bagel and an O-ring. A ringette ring is torus-shaped.
A torus is like a tube that is bent into a circle so it connects to itself. The radius of the tube or circle is called the minor radius, written as [math]\displaystyle{ r }[/math]. The distance from the center of the tube to the center of the torus is called the major radius, written as [math]\displaystyle{ R }[/math].
The surface area of a torus is given by
- [math]\displaystyle{ \begin{align} A &= \left( 2\pi r \right) \left(2 \pi R \right) = 4 \pi^2 R r \end{align} }[/math].
This area is the same as the area of a straight tube that has a radius [math]\displaystyle{ r }[/math] and length [math]\displaystyle{ 2\pi R }[/math].
The volume of a solid torus is given by
- [math]\displaystyle{ \begin{align} V &= \left ( \pi r ^2 \right ) \left( 2 \pi R \right) = 2 \pi^2 R r^2 \end{align} }[/math].
This volume is the same as the volume of a straight rod that has a radius [math]\displaystyle{ r }[/math] and length [math]\displaystyle{ 2\pi R }[/math].
Torus Media
A ring torus with a selection of circles on its surface
As the distance from the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a double-covered sphere.
A ring torus with aspect ratio 3, the ratio between the diameters of the larger (magenta) circle and the smaller (red) circle.
Poloidal direction (red arrow) and toroidal direction (blue arrow)
Turning a punctured torus inside-out
A stereographic projection of a Clifford torus in four dimensions performing a simple rotation through the xz-plane
The configuration space of 2 not necessarily distinct points on the circle is the orbifold quotient of the 2-torus, T2 / S2, which is the Möbius strip.
The Tonnetz is an example of a torus in music theory.The Tonnetz is only truly a torus if enharmonic equivalence is assumed, so that the (F♯-A♯) segment of the right edge of the repeated parallelogram is identified with the (G♭-B♭) segment of the left edge.
In three dimensions, one can bend a rectangle into a torus, but doing this necessarily affects the distances measured along the surface, as seen by the distortion of the checkered pattern.
Seen in stereographic projection, a 4D flat torus can be projected into 3-dimensions and rotated on a fixed axis.
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